Possible Charged Particle Field
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Re: Possible Charged Particle Field
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I'll share my meager progress.
Jump right into overdrive. EQS406. With r=1 particles, the minimum spherical radius  without particle contact, is just under 14. At that distance, the r=1 particles are 8.35 angular degrees. Contact occurs between the 3 and the nine particle ring latitudes. Using equatorial offsets, we can add an equator and a new latitude ring. The original 8 equatorial latitude rings, (with 30,30,36,36,36,36,30,30 particles) separated by about 10 degrees have been replaced with 11 latitude rings (of 30 particles each) separated by 7.1916 degrees. N increased from 406 to 472. Not shown. That number could be increased further by addressing the pole latitudes, but I don’t see the point pushing past overdrive – that’s supposed to be our limit. It makes sense to start with a smaller EQS solution, with two less latitude rows compared with 406, then optimize that solution to replace the existing EQS406.
With that in mind I’ll begin with EQS372 northern hemisphere latitudes (up from the equator at zero) and particle ring counts.
[ 11.25, 12 ], // 30 particles
[ 22.5, 12 ], // 30 "
[ 33.75, 12 ], // 30 "
[ 45, 17.1429 ], // 21 "
[ 56.25, 20 ], // 18 "
[ 67.5, 24 ], // 15 "
[ 78.75, 40 ], // 9 "
[ 88.5, 120 ] // 3 "
Working the Overdrive EQS optimization.
Using equatorial offsets, of 360/30/2 degrees, I added a new 30 particle equatorial ring which bumped the EQS372 up to a modEQS402. With r=1 particles, the minimum spherical particle configuration radius  without particle contact, for the 372 (and 402) is just under 12, two smaller than the 406 radius, so the particles appear larger in the modEQS402 image.
I see the 402 is a good improvement in particle uniformity over the EQS406. I'll work the pole ends next, beginning with rotating the 3 particle pole rings by 20 degrees about the poletopole axis. I'll move and may even might add a new polar latitude as well. All changes are straightforward. The particle count should increase a little bit more.
I'll be busy offline for the next two days.
.
I'll share my meager progress.
Jump right into overdrive. EQS406. With r=1 particles, the minimum spherical radius  without particle contact, is just under 14. At that distance, the r=1 particles are 8.35 angular degrees. Contact occurs between the 3 and the nine particle ring latitudes. Using equatorial offsets, we can add an equator and a new latitude ring. The original 8 equatorial latitude rings, (with 30,30,36,36,36,36,30,30 particles) separated by about 10 degrees have been replaced with 11 latitude rings (of 30 particles each) separated by 7.1916 degrees. N increased from 406 to 472. Not shown. That number could be increased further by addressing the pole latitudes, but I don’t see the point pushing past overdrive – that’s supposed to be our limit. It makes sense to start with a smaller EQS solution, with two less latitude rows compared with 406, then optimize that solution to replace the existing EQS406.
With that in mind I’ll begin with EQS372 northern hemisphere latitudes (up from the equator at zero) and particle ring counts.
[ 11.25, 12 ], // 30 particles
[ 22.5, 12 ], // 30 "
[ 33.75, 12 ], // 30 "
[ 45, 17.1429 ], // 21 "
[ 56.25, 20 ], // 18 "
[ 67.5, 24 ], // 15 "
[ 78.75, 40 ], // 9 "
[ 88.5, 120 ] // 3 "
Working the Overdrive EQS optimization.
Using equatorial offsets, of 360/30/2 degrees, I added a new 30 particle equatorial ring which bumped the EQS372 up to a modEQS402. With r=1 particles, the minimum spherical particle configuration radius  without particle contact, for the 372 (and 402) is just under 12, two smaller than the 406 radius, so the particles appear larger in the modEQS402 image.
I see the 402 is a good improvement in particle uniformity over the EQS406. I'll work the pole ends next, beginning with rotating the 3 particle pole rings by 20 degrees about the poletopole axis. I'll move and may even might add a new polar latitude as well. All changes are straightforward. The particle count should increase a little bit more.
I'll be busy offline for the next two days.
.
LongtimeAirman Admin
 Posts : 1196
Join date : 20140810
Re: Possible Charged Particle Field
.
EQS406 versus modEQS372.
Our simulated charge field particles react to the ambient charge field according to the ‘charge’ received at their charge sampling points. Those points are currently calculated by an equiisolatitudinal area algorithm called EQS. We currently use three EQS solutions  46, 130 and 406.
Any algorithm has its purpose and limitations. We may improve the behavior of our charge field particles by increasing the uniformity of the sample point distribution sets over our current EQS precision ‘standards’ by careful modifications. We can conveniently display those sampling point distribution sets by placing neutral particles in the ‘same’ spherical configurations for easy review. These mods may indicate simple improvements to the algorithm itself.
EQS406 compared to EQS 342 modified into modEQS372.
Please reference the above image. In my last post I showed the ‘Overdrive’ spherical configuration  EQS406 and possible alternative configuration, modEQS402 still needing some pole optimization. That pole mod led me to begin at a different EQS solution, modifying EQS342 into a modEQS372, currently my recommended replacement for EQS406.
As in my previous post, with r=1 particles, the minimum spherical particle configuration radius without particle contact, for the EQS342 (and modEQS372) is just under 12. The ‘minimum’ EQS406 radius without particle contact is 14. The EQS406 has 9 northern hemisphere latitude particle rings separated by about 10 degrees: (3, 9, 15, 20, 24, 30, 30, 36, 36). That same set is repeated in the southern hemisphere.
Starting with EQS342  (3, 9, 15, 18, 21, 30, 30, 30, 30, 30, 30, 21, 18, 15, 9, 3). Sixteen latitude rings, totaling 342 points. Make the following modifications:
1. Adding single particles over the poles. EQS doesn’t provide single pole particles. [ 90, 360 ] results in a particle that ‘leans’ on the ptop axis.
2. Offsetting adjacent pole latitude rings. EQS partitions the spherical surface in both latitude and longitudinal alignments. Instead of the current six minimum gaps; point the hex between 2 of the 12 particles in the next particle ring instead of directly at six of them by rotating the 6 particle ring latitude 360/12/2 or 15 degrees. If the pole was three point, it would need the same rotation with respect to its neighboring latitude. Pole latitudes numbers that aren't evenly divisible may not benifit from offset rotations. Note, the modEQS372 in the image still needs its hex rotation. EQS342’s polar latitude group: 3, 9, 15, 18, 21 was changed to modEQS372’s 1, 6, 12, 18, 21, 24.
3. Offsetting adjacent equatorial latitude rings. Given rows of equal number (30) particle rings, we can increase the distance between adjacent (elevation) particles by alternating the starting positions of the each particle ring 360/30/2, 6 degrees. Offsetting adjacent latitudes allows us to change reduce the elevation angle of adjacent latitudes results the resulting distinctive hexagonal alignments.
4. Adding a new 30 particle ring over the equator. Note, all EQS algorithm solutions lack equatorial rings. Offsetting the equatorial latitudes has allowed us to replace the previous 6 latitudes with 10 degree separation to 7 latitudes, each separated by 8.7 degrees, with resulting distinctive hexagonal alignments.
The result is modEQS372  (1, 6, 12, 18, 21, 24, 30, 30, 30, 30, 30, 30, 30, 24, 21, 18, 12, 6, 1). 19 latitudes with 372 points. The modifications above result in increased numbers of particles at EQS342’s radius of 12, without allowing r=1 particles to overlap; EQS406’s minimum radius without r=1 particle overlap is 14.
The new modified distribution is densest between adjacent (azimuthal) particles within the two +/30 particle ring latitudes at the shoulders. The ‘density’ of points reduces toward both the equator and poles.
The main limitation of this method is maximizing the number of equal particle ring latitudes. I believe there are just a small set of two modifiable EQS solutions at this ‘scale’, involving only those solutions with 4 or 6 equal particle length equatorial latitudes. ModEQS372 handles the 6 case, six equatorial latitudes become the modified seven. The EQS130 competition will handle the other, by changing an EQS solution with 4 adjacent (elevation) equal particle equatorial latitudes into the modified five.
EQS130 contenders next.
.
EQS406 versus modEQS372.
Our simulated charge field particles react to the ambient charge field according to the ‘charge’ received at their charge sampling points. Those points are currently calculated by an equiisolatitudinal area algorithm called EQS. We currently use three EQS solutions  46, 130 and 406.
Any algorithm has its purpose and limitations. We may improve the behavior of our charge field particles by increasing the uniformity of the sample point distribution sets over our current EQS precision ‘standards’ by careful modifications. We can conveniently display those sampling point distribution sets by placing neutral particles in the ‘same’ spherical configurations for easy review. These mods may indicate simple improvements to the algorithm itself.
EQS406 compared to EQS 342 modified into modEQS372.
Please reference the above image. In my last post I showed the ‘Overdrive’ spherical configuration  EQS406 and possible alternative configuration, modEQS402 still needing some pole optimization. That pole mod led me to begin at a different EQS solution, modifying EQS342 into a modEQS372, currently my recommended replacement for EQS406.
As in my previous post, with r=1 particles, the minimum spherical particle configuration radius without particle contact, for the EQS342 (and modEQS372) is just under 12. The ‘minimum’ EQS406 radius without particle contact is 14. The EQS406 has 9 northern hemisphere latitude particle rings separated by about 10 degrees: (3, 9, 15, 20, 24, 30, 30, 36, 36). That same set is repeated in the southern hemisphere.
Starting with EQS342  (3, 9, 15, 18, 21, 30, 30, 30, 30, 30, 30, 21, 18, 15, 9, 3). Sixteen latitude rings, totaling 342 points. Make the following modifications:
1. Adding single particles over the poles. EQS doesn’t provide single pole particles. [ 90, 360 ] results in a particle that ‘leans’ on the ptop axis.
2. Offsetting adjacent pole latitude rings. EQS partitions the spherical surface in both latitude and longitudinal alignments. Instead of the current six minimum gaps; point the hex between 2 of the 12 particles in the next particle ring instead of directly at six of them by rotating the 6 particle ring latitude 360/12/2 or 15 degrees. If the pole was three point, it would need the same rotation with respect to its neighboring latitude. Pole latitudes numbers that aren't evenly divisible may not benifit from offset rotations. Note, the modEQS372 in the image still needs its hex rotation. EQS342’s polar latitude group: 3, 9, 15, 18, 21 was changed to modEQS372’s 1, 6, 12, 18, 21, 24.
3. Offsetting adjacent equatorial latitude rings. Given rows of equal number (30) particle rings, we can increase the distance between adjacent (elevation) particles by alternating the starting positions of the each particle ring 360/30/2, 6 degrees. Offsetting adjacent latitudes allows us to change reduce the elevation angle of adjacent latitudes results the resulting distinctive hexagonal alignments.
4. Adding a new 30 particle ring over the equator. Note, all EQS algorithm solutions lack equatorial rings. Offsetting the equatorial latitudes has allowed us to replace the previous 6 latitudes with 10 degree separation to 7 latitudes, each separated by 8.7 degrees, with resulting distinctive hexagonal alignments.
The result is modEQS372  (1, 6, 12, 18, 21, 24, 30, 30, 30, 30, 30, 30, 30, 24, 21, 18, 12, 6, 1). 19 latitudes with 372 points. The modifications above result in increased numbers of particles at EQS342’s radius of 12, without allowing r=1 particles to overlap; EQS406’s minimum radius without r=1 particle overlap is 14.
The new modified distribution is densest between adjacent (azimuthal) particles within the two +/30 particle ring latitudes at the shoulders. The ‘density’ of points reduces toward both the equator and poles.
The main limitation of this method is maximizing the number of equal particle ring latitudes. I believe there are just a small set of two modifiable EQS solutions at this ‘scale’, involving only those solutions with 4 or 6 equal particle length equatorial latitudes. ModEQS372 handles the 6 case, six equatorial latitudes become the modified seven. The EQS130 competition will handle the other, by changing an EQS solution with 4 adjacent (elevation) equal particle equatorial latitudes into the modified five.
EQS130 contenders next.
.
Last edited by LongtimeAirman on Thu Feb 21, 2019 7:00 pm; edited 1 time in total (Reason for editing : Typos!)
LongtimeAirman Admin
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Re: Possible Charged Particle Field
.
EQS130, along with two alternatives, modEQS128 and modEQS144 are shown. A few small changes to the existing EQS algorithm (ex. EQS 130) result in the two modified and, I believe, improved (equidistant) point distributions  as shown.
You have my sympathies, post after post of particles in spherical configurations. This is a working project and the devil’s in the details. As you no doubt already know (too well), the present goal is to improve the uniformity of distribution (equidistance wise) of our calculated particle’s charge field sampling points. That should have the side benefit of improved overall performance of our possible charged particle field simulation engine.
Note that the engine is being used as a tool to model the sampling points of a single particle as particle configurations. Aside from my autocad modifications, our particle engine allows one to make these models. I consider them as real and valid in some sense, although to be perfectly honest, I cannot explain nor justify the validity of spherical particle configurations with respect to the charge field. Nevertheless, I’ll work toward developing a spherical group UI that will allow a casual user to create many of them herself. Then that single spherical UI scenario could allow me to discard several redundant and tool scenarios currently in the Spherical group.
The user just selects from these precision levels: Low, Med, High and Overdrive. The default is low. I discussed Overdrive in my last post. EQS130 is our simulation’s high Precision standard, made up of (3, 9, 15, 18, 20, 20, 18, 15, 9, 3), 10 particle ring latitudes totaling 130 particles, and is shown in the top row of the image. This configuration begins to overlap the r=1 particles at about R=7.1. R=7.5 in the EQS130 first row of images.
ModEQS128. – (1, 6, 12, 18, 18, 18, 18, 18, 12, 6, 1), 11 rings, 128 particles. The first alternative to ECP130 begins with EQS108 – (6, 12, 18, 18, 18, 18, 12, 6), 10 particle ring latitudes totaling 108 particles. The configuration begins to overlap the r=1 surface particles near R=6.6. R=7 in the EQS108, as well as the modEQS128– second row images. The modifications are as follows:
1. Add single pole particles.
2. Rotate the 6 particle ring pole, 360/12/2 = 15 degrees. This allows slightly more room between the 6 and 12 particle ring latitudes – say for a slightly larger 6, or smaller 12 particle rings.
3. Convert the four equatorial 18 particle rings into five by adding a new 18 particle ring and
4. Offsetting, or alternating the starting point of each adjacent 18 particle latitude ring by 360/18/2 = 10 degrees (az) so that the 5 rows will form a hexagonal pattern.
ModEQS144 – (3, 9, 15, 18, 18, 18, 18, 18, 15, 9, 3), 11 rings, 144 particles. It is also shown (third row) at the configuration radius R=7. It began as EQS126 (3, 9, 15, 18, 18, 18, 18, 15, 9, 3). Here are the 144 modifications:
1. Rotate the pole triangles 360/9/2 = 20 degrees (az).
2. Offset equal particle (18) equatorial latitudes 360/18/2 = 10 degrees, thereby making room to;
3. Add a new 18 particle latitude ring, increasing the number equatorial latitudes from 4 to 5 in the same area.
I believe both the ModEQS128 and ModEQS144 alternatives provide more uniform point distributions then EQS130.
I’ll look at EQS46, the Medium Precision, next.
.
EQS130, along with two alternatives, modEQS128 and modEQS144 are shown. A few small changes to the existing EQS algorithm (ex. EQS 130) result in the two modified and, I believe, improved (equidistant) point distributions  as shown.
You have my sympathies, post after post of particles in spherical configurations. This is a working project and the devil’s in the details. As you no doubt already know (too well), the present goal is to improve the uniformity of distribution (equidistance wise) of our calculated particle’s charge field sampling points. That should have the side benefit of improved overall performance of our possible charged particle field simulation engine.
Note that the engine is being used as a tool to model the sampling points of a single particle as particle configurations. Aside from my autocad modifications, our particle engine allows one to make these models. I consider them as real and valid in some sense, although to be perfectly honest, I cannot explain nor justify the validity of spherical particle configurations with respect to the charge field. Nevertheless, I’ll work toward developing a spherical group UI that will allow a casual user to create many of them herself. Then that single spherical UI scenario could allow me to discard several redundant and tool scenarios currently in the Spherical group.
The user just selects from these precision levels: Low, Med, High and Overdrive. The default is low. I discussed Overdrive in my last post. EQS130 is our simulation’s high Precision standard, made up of (3, 9, 15, 18, 20, 20, 18, 15, 9, 3), 10 particle ring latitudes totaling 130 particles, and is shown in the top row of the image. This configuration begins to overlap the r=1 particles at about R=7.1. R=7.5 in the EQS130 first row of images.
ModEQS128. – (1, 6, 12, 18, 18, 18, 18, 18, 12, 6, 1), 11 rings, 128 particles. The first alternative to ECP130 begins with EQS108 – (6, 12, 18, 18, 18, 18, 12, 6), 10 particle ring latitudes totaling 108 particles. The configuration begins to overlap the r=1 surface particles near R=6.6. R=7 in the EQS108, as well as the modEQS128– second row images. The modifications are as follows:
1. Add single pole particles.
2. Rotate the 6 particle ring pole, 360/12/2 = 15 degrees. This allows slightly more room between the 6 and 12 particle ring latitudes – say for a slightly larger 6, or smaller 12 particle rings.
3. Convert the four equatorial 18 particle rings into five by adding a new 18 particle ring and
4. Offsetting, or alternating the starting point of each adjacent 18 particle latitude ring by 360/18/2 = 10 degrees (az) so that the 5 rows will form a hexagonal pattern.
ModEQS144 – (3, 9, 15, 18, 18, 18, 18, 18, 15, 9, 3), 11 rings, 144 particles. It is also shown (third row) at the configuration radius R=7. It began as EQS126 (3, 9, 15, 18, 18, 18, 18, 15, 9, 3). Here are the 144 modifications:
1. Rotate the pole triangles 360/9/2 = 20 degrees (az).
2. Offset equal particle (18) equatorial latitudes 360/18/2 = 10 degrees, thereby making room to;
3. Add a new 18 particle latitude ring, increasing the number equatorial latitudes from 4 to 5 in the same area.
I believe both the ModEQS128 and ModEQS144 alternatives provide more uniform point distributions then EQS130.
I’ll look at EQS46, the Medium Precision, next.
.
LongtimeAirman Admin
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Re: Possible Charged Particle Field
How is this going, Airman? Any new advancements?
Nevyn Admin
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Re: Possible Charged Particle Field
.
I don’t see how the methodology I’ve been using can “optimize” the EQS46. I suppose I could come up with an objective quantitative measure of equispaced points in order to evaluate the EQS46, and then compare it to an alternative, there are many standard polyhedral forms to choose from. I also wanted to give an additional look at the geodesics. Also I’m sure the ‘fullerene’ icosabased carbon configuration, C60 would be better choice than the EQS46.
Here’s another idea I’ve picked up in my readings, that I need to pass by you. Particle placements based on repulsion; our possible charged particle field should be able to handle that. Using the EQP algorithm, an arbitrary number of Protons can be positioned at the spherical configuration’s radius with their south poles oriented toward the spherical configuration’s center. I imagine the protons’ emission fields may then push each proton into optimal positions which we may then use as an ideal Precision solution set for that number of points.
In order to do that, we must constrain the protons’ motions to the spherical radius. Specifically, is it possible  relatively easily – to: 1. Immobilize, fix, or make Unmoveable  both the proton’s spin axis to always point to the configuration center; as well as 2. Fix the protons distance from that center?
Optimistic, eh? I had to ask.
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I don’t see how the methodology I’ve been using can “optimize” the EQS46. I suppose I could come up with an objective quantitative measure of equispaced points in order to evaluate the EQS46, and then compare it to an alternative, there are many standard polyhedral forms to choose from. I also wanted to give an additional look at the geodesics. Also I’m sure the ‘fullerene’ icosabased carbon configuration, C60 would be better choice than the EQS46.
Here’s another idea I’ve picked up in my readings, that I need to pass by you. Particle placements based on repulsion; our possible charged particle field should be able to handle that. Using the EQP algorithm, an arbitrary number of Protons can be positioned at the spherical configuration’s radius with their south poles oriented toward the spherical configuration’s center. I imagine the protons’ emission fields may then push each proton into optimal positions which we may then use as an ideal Precision solution set for that number of points.
In order to do that, we must constrain the protons’ motions to the spherical radius. Specifically, is it possible  relatively easily – to: 1. Immobilize, fix, or make Unmoveable  both the proton’s spin axis to always point to the configuration center; as well as 2. Fix the protons distance from that center?
Optimistic, eh? I had to ask.
.
LongtimeAirman Admin
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Re: Possible Charged Particle Field
Yes, those things are possible, but it isn't pretty and breaks the model we are trying to create. It would require a custom model. I don't think it is worth the effort. It also won't work because the emission gives the particles a velocity which doesn't stop when it reaches some optimal position. They will just keep going (you could step through it, frame by frame to fix this). Plus, how are you going to place them at the start? Wouldn't that placement choose the final positions, with slight variations given that the protons emission is not spherical? That emission profile is a problem too. We don't want it to be a factor is choosing charge reception, only emission.
In order to test the charge point configurations, we can use a bit of math. There is probably some better way to do it, but when I'm in unknown territory and can't see a nice solution, I tend to brute force it and then look for optimizations later.
We take each generated charge point and we calculate the distance to all other particles. We then find the minimum value from that set. So we end up with a minimum value per charge point. Then we compare all of the minimum values to each other. There are many ways to do that but just putting them into a graph would be a good start. If you end up with a straight horizontal line, then you have found an optimal solution. The most optimal solution is that which is closest to Y=0 on the graph.
In order to test the charge point configurations, we can use a bit of math. There is probably some better way to do it, but when I'm in unknown territory and can't see a nice solution, I tend to brute force it and then look for optimizations later.
We take each generated charge point and we calculate the distance to all other particles. We then find the minimum value from that set. So we end up with a minimum value per charge point. Then we compare all of the minimum values to each other. There are many ways to do that but just putting them into a graph would be a good start. If you end up with a straight horizontal line, then you have found an optimal solution. The most optimal solution is that which is closest to Y=0 on the graph.
Nevyn Admin
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Re: Possible Charged Particle Field
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With respect to points on a sphere, I believe repulsion is actually an iterative process. Based on increasing the distance between the closest points in the set of N points on a sphere at a given iteration. It’s another possible alternative to our current EQS and EQP algorithms I hadn’t gotten into  there are many. Equidistant points on a sphere aren't true beyond the platonic solids, and are only approximated. I thought if we had ‘real proton repulsion’  given an EQP starting set – it might work; but you’re correct, I ‘overlooked’ the constant motion of the nonzero velocities as the protons keep repelling each other.
I should point out my obvious thinking. Fixing orientations and distances in the way I suggested mimics the coherence orientation. I suppose any coherence in our possible charged particle field would also require a custom model.
With that in mind, and not for the first time, I googled something like “equispacing points on a sphere”, and found an algorithm based on scurves on the sphere. It’s undergraduate, easy to read for anyone interested in the subject. Nice looking results and it also includes an efficiency function that sounds like what you had in mind.
A New Computationally Efficient Method for Spacing n Points on a Sphere Jonathan Kogan https://scholar.rosehulman.edu/rhumj/vol18/iss2/5/
Pages 54 – 71.
Also, the results look good. The Mathematica code is included, here's the Python code.
Airman. Ok, nix the repulsion idea.Nevyn wrote: Yes, those things are possible, but it isn't pretty and breaks the model we are trying to create. … .
With respect to points on a sphere, I believe repulsion is actually an iterative process. Based on increasing the distance between the closest points in the set of N points on a sphere at a given iteration. It’s another possible alternative to our current EQS and EQP algorithms I hadn’t gotten into  there are many. Equidistant points on a sphere aren't true beyond the platonic solids, and are only approximated. I thought if we had ‘real proton repulsion’  given an EQP starting set – it might work; but you’re correct, I ‘overlooked’ the constant motion of the nonzero velocities as the protons keep repelling each other.
I should point out my obvious thinking. Fixing orientations and distances in the way I suggested mimics the coherence orientation. I suppose any coherence in our possible charged particle field would also require a custom model.
I take it you would like to see some quantitative measure in support of selecting alternative precision candidates, suggesting “brute force” for a start. That’s a change, up to now, I’ve been relying on images to compare the precision candidates, I’ll shoot for numbers too.Nevyn wrote: In order to test the charge point configurations. ...
We take each generated charge point and we calculate the distance to all other particles. …
With that in mind, and not for the first time, I googled something like “equispacing points on a sphere”, and found an algorithm based on scurves on the sphere. It’s undergraduate, easy to read for anyone interested in the subject. Nice looking results and it also includes an efficiency function that sounds like what you had in mind.
A New Computationally Efficient Method for Spacing n Points on a Sphere Jonathan Kogan https://scholar.rosehulman.edu/rhumj/vol18/iss2/5/
Pages 54 – 71.
Pg 62. … we ﬁrst created three new functions to analyze how well the points were spaced on a sphere: SmallestDistance, TheoreticalSmallestDistance, and NormalizedSmallestDistance.
SmallestDistance(pts) takes a list of points and identiﬁes the smallest Euclidean distance between any two points on the sphere. In other words, it identiﬁes the two points that are closest together and outputs the distance between the two.
The TheoreticalSmallestDistance(n) function returns the upper bound for the distance of n equally spaced points on a sphere. It is well known that the optimal spacing of points on the plane is the hexagonal grid. To ﬁnd the upper bound for the distance of n points on a sphere, we imagine there is a planar hexagonal grid that covers the entire area of the sphere. Using this grid, we ﬁnd this upper bound for all n values based on the size and the number of equilateral triangles.
Lastly, we deﬁned the NormalizedSmallestDistance(pts) function as the SmallestDistance(pts)/TheoreticalSmallestDistance(Length(pts)); the function provides a relative accuracy for the points that are around the sphere.
Also, the results look good. The Mathematica code is included, here's the Python code.
Pg 70. The research discussed in this paper has also led to the creation of the Mathematica function SpherePoints[n], which was added in the Mathematica 11.1 update [10]. This Mathematica function enables anybody who wants points spaced equally on a sphere to generate them quickly and easily [8].
 Code:
Pg 72. A.2 Python Code
import math
def spherical coordinate (x , y ):
return [math. cos (x) ∗ math. cos (y) ,
math. sin (x) ∗ math. cos (y) , math. sin (y )]
def NX(n, x ):
pts =[]
start = (−1. + 1. / (n − 1.))
increment = (2. − 2. / (n − 1.)) / (n − 1.)
for j in xrange(0 , n):
s = start + j ∗ increment
pts . append(
spherical coordinate (
s ∗ x , math. pi / 2. ∗
math. copysign (1 , s ) ∗
(1. − math. sqrt (1. − abs( s )))
))
return pts
def generate points (n):
return NX(n, 0.1 + 1.2 ∗ n)
LongtimeAirman Admin
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Re: Possible Charged Particle Field
.
Status update. Unfortunately, Jonathan Kogan’s paper didn’t provide all the details. The program is more complicated than it first appeared, and I haven’t decided how I should go about reorganizing the code I previously posted. The important thing is supposed to be the “efficiency measure”. Requoting TheoreticalSmallestDistance(n), “we imagine there is a planar hexagonal grid that covers the entire area of the sphere”, that’s all the information provided on that subject.
I decided to go through the exercise of figuring out the efficiency measure while adding a simpler type of spiral algorithm, based on the Golden ratio. From, Evenly distributed points on sphere,
https://web.archive.org/web/20060728050730/http://cgafaq.info/wiki/Evenly_Distributed_Points_On_Sphere. Also Points on a sphere, October 4th, 2006 by Patrick Boucher. //http://www.softimageblog.com/archives/115
In the meantime, I found Jonathan Kogan’s youtube video of his paper, A New Computationally Efficient Method for Spacing n Points on a Sphere
.
The Efficiency measure, including the TheoreticalSmallestDistance(n) equation I was searching for appears to be part of a small set (3 or 4(?)) of slides – not included in the paper  that served for brief glimpses in the two videos, good enough for a screen capture.
Outputs for the new Golden ratio spherical algorithm with Efficiency measure are shown. Above, the spherical configuration GS50 is on the left and GS406 (from the inside) is on the right. We are looking down along the main N/S y axis for both. While the overall spacing is nice, the poles are either clumped or open.
The Efficiency Measure for GS406 is shown below. Assuming I implemented it correctly the normalized efficiency measure is a poor 0.674. According to Kogan’s paper the best efficiences are about 0.85. Radius 16 is cutting it close, the two closest r=1 neutrons are separated by just 0.040 – almost touching. If one were interested in the just the efficiency numbers, it’s easy to switch between the Parameters and Output control tabs. Curious to see the results, beginning with 10, and incrementing by five up to 65, then 130, then 406, the best efficiency I found was 0.7115 for 10 neutrons, with a fairly straight line dropping to the the GS406’s 0.674. The Golden ratio I implemented is not a good source for equidistant spaced points necessary for our desired charge detection Precision candidates.
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Status update. Unfortunately, Jonathan Kogan’s paper didn’t provide all the details. The program is more complicated than it first appeared, and I haven’t decided how I should go about reorganizing the code I previously posted. The important thing is supposed to be the “efficiency measure”. Requoting TheoreticalSmallestDistance(n), “we imagine there is a planar hexagonal grid that covers the entire area of the sphere”, that’s all the information provided on that subject.
I decided to go through the exercise of figuring out the efficiency measure while adding a simpler type of spiral algorithm, based on the Golden ratio. From, Evenly distributed points on sphere,
https://web.archive.org/web/20060728050730/http://cgafaq.info/wiki/Evenly_Distributed_Points_On_Sphere. Also Points on a sphere, October 4th, 2006 by Patrick Boucher. //http://www.softimageblog.com/archives/115
 Code:
inc := pi*(3sqrt(5))
off := 1/(2*N)  1
for k := 0 .. N1
z := k/N + off
r := sqrt(1z*z)
phi := k*inc
pt[k] := (cos(phi)*r, sin(phi)*r, z)
In the meantime, I found Jonathan Kogan’s youtube video of his paper, A New Computationally Efficient Method for Spacing n Points on a Sphere
.
The Efficiency measure, including the TheoreticalSmallestDistance(n) equation I was searching for appears to be part of a small set (3 or 4(?)) of slides – not included in the paper  that served for brief glimpses in the two videos, good enough for a screen capture.
 Code:
smallestDistance[pts_] := min[table[N[euclideanDistance[pts[[i]], pts[[j]]]],{i,1.0, length[pts]}, {j, i + 1.0, length[pts]}]]
theorecticalSmallestDistance[pts_] := Math.sqrt(8*Math.PI/(n*Math.sqrt(3)))
normalizedSmallestDistance[pts_]:= smallestDistance[pts]/theorecticalsmallestDistance[length[pts_]]
Outputs for the new Golden ratio spherical algorithm with Efficiency measure are shown. Above, the spherical configuration GS50 is on the left and GS406 (from the inside) is on the right. We are looking down along the main N/S y axis for both. While the overall spacing is nice, the poles are either clumped or open.
The Efficiency Measure for GS406 is shown below. Assuming I implemented it correctly the normalized efficiency measure is a poor 0.674. According to Kogan’s paper the best efficiences are about 0.85. Radius 16 is cutting it close, the two closest r=1 neutrons are separated by just 0.040 – almost touching. If one were interested in the just the efficiency numbers, it’s easy to switch between the Parameters and Output control tabs. Curious to see the results, beginning with 10, and incrementing by five up to 65, then 130, then 406, the best efficiency I found was 0.7115 for 10 neutrons, with a fairly straight line dropping to the the GS406’s 0.674. The Golden ratio I implemented is not a good source for equidistant spaced points necessary for our desired charge detection Precision candidates.
.
Last edited by LongtimeAirman on Fri Mar 15, 2019 7:05 pm; edited 1 time in total (Reason for editing : Added efficiency code)
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